Last part of question on continuous functions

[nlews]

Homework Statement

This is the last part of a revision question I'm trying, would really like to get to the end so any pointers or help would be greatly appreciated.

Suppose h:(0,1)-> satisfies the following conditions:
for all xЭ(0,1) there exists d>0 s.t. for all x'Э(x, x+d)n(0,1) we have h(x)<=h(x')

Prove that if h is continuous on (0,1) then h(x)<=h(y) whenever x.yЭ(0,1) and x<=y. Use a counterexample to show that this results may not be true when h is continuous.

Homework Equations

Well, definition of continuous functions,
also above in the question i am asked to state the intermediate value theorem so i think perhaps i am meant to use that.

The Attempt at a Solution

if h is continuous on (0,1) then for all cЭ(0,1), for all E>0, Эd s.t. for all xЭ(0,1), 0<|x-c|<d this implies that |h(x)-h(c)|<E
i really just don't see how to continue. Any help would be great. Thank you

 

[Office_Shredder]

Try fixing x, and showing that for every value of y larger than it, h(x)<=h(y)